RF antenna to measure polarization

Is it possible to build a narrow-band antenna (or antenna+circuit) to measure the RF (or microwave) polarization?

Or, say, which amplitude is proportional to cosine of some angle between E and the antenna (however, I need 3 such antennas then)

To measure antenna polarization you can use a half-wave dipole or a waveguide horn. Their typical polarization separation is >30 dB if made geometrically "clan". Depends on frequency what you may prefer.
At microwave bands I used dipoles made of a thin metal sheet, to achieve >30 dB polarization ratio.
Locate this test antenna far enough from the tested antenna, on a common line, then turn one of the antennas on this axis. Record the maximum and minimum; their ratio is the polarization ratio of the antenna pair.

To achieve >>30 dB, it is recommended to use an additional grid made of parallel wires and located before the test antenna. The grid "purifies" the wave front by ~20 dB in polarization ratio.